I've totally done this. During 50 States, at the Eastern Market stop I realized I didn't have cash for my favorite pretzels (the guy who sets up in front of the the pool entrance), so I rode through the drive-through ATM at 6th/C SE, since there isn't a walk-up machine.

I've biked up to 'drive-through' tellers several times over the years with no problems. Even walked up to my own bank 'drive-through' a few times with no fuss when their indoor counters have closed. I always feel a sense of accomplishment and pride that I don't need a car to use a 'drive-through' (although I realize that is essentially why they exist). However, I'm usually careful not to cue on bike/foot when there's a line of cars waiting.